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Pound-Star Notation
Pound-Star Notation is an in-progress notation by SuperJedi224.Almost Infinite The latest revision of the notation, Multi-Nested Hyper-Exploding Pound-Star (H#*<<>>), is defined as follows: An expression consists of one or more entries separated by groups consisting of one or more items, each of which may be either a star character (*), a carat (^), an explodon array, denoted as a colon-separated list of nonnegative integers, of which at least one must be positive, wrapped in curly brackets; an at sign (@), a nested group (denoted as any valid separator group- even one containing other nesting groups or supernests- wrapped in angle brackets), or a supernest, denoted as either a nonnegative integer or any valid separator group in square brackets or an empty pair of square brackets. The first entry must be a pound sign (#). Subsequent entries my be either a pound sign, a single positive integer, an ordered, comma-separated, set of one or more nonnegative integers, of which at least one must be positive, wrapped in a single set of parentheses; or a "proto-set" denoted as a single positive integer wrapped in two pairs of parentheses. For instance, one valid expression is #@{0:3}((2)){2}(1,3){2}2**#*7. Rules # Start with n=1. # If the final separator group is anything other than a single *: ## If the final item in the group is a supernest: ### If it is empty, put the current value of n in it ### If it contains the number 0, replace it with an at sign ### If it contains any other number, decrement the number by one and wrap the supernest in n layers of nesting groups ### If it contains a separator group: #### If the last item in the supernest is a star character, remove it and wrap the supernest in n layers of nesting groups #### Otherwise, reduce it as you would a normal separator group ## If the final item in the group is any nesting group, replace it with n concatenated copies of whatever subgroup it contains ## If the final item in the group is an at sign, replace it with an explodon array of n terms, of which the last term is 1 and all preceding terms are 0. ## If the final item in the group is an explodon array: ### If there are two or more terms, of which the last is 0, remove the last term. ### Else, if the first term is 0: ####If that is the only term, replace that explodon array with a carat. ####Else, decrement the first nonzero term by one and set all earlier terms to n. ### Else, replace that explodon array with n identical explodon arrays that have had the first term decremented by 1. ## If the final character in the group is a carat, replace it with n stars. ## If the final character in the group is a star, replace the final entry with a series of n identical entries whose separator groups have had the final star removed. # Else: ## If the last entry is a number, multiply n by this number, then remove it from the expression. ## If the last entry is a pound sign, replace it with the current n. ## If the last entry is a proto-set ((x)), replace it with a set of n elements each equalling x. ## If the last entry is a set: ### If the set contains only one element, replace the set with that number raised to the power of the current n. ### Else, if the final element is 0, remove it and increment n by 1. ### Else, decrement the final element by 1 and increment the element preceding it by n. # Repeat rules 2-3 until the expression is reduced to a single number. This is the value of the expression. Growth Rate The latest extension has a limit ordinal of about \(\varepsilon_0\) in the fast-growing hierarchy. Low-level example #*(1,0,1)*2 n=1, #*(1,0,1)*2 n=2, #*(1,0,1) n=2, #*(1,2,0) n=3, #*(1,2) n=3, #*(4,1) n=3, #*(7,0) n=4, #*(7) n=4, #*2,401 n=9,604, # 9,604 Sources Category:Notations